Here, \(\delta >0\)  is added to make the square root function smooth. In particular, we consider a semi-parametric data generating process given by a sparse single-index model of the form \(b = g(\langle a, x^* \rangle ) +\zeta \) , where \(g:\mathbb {R} \rightarrow \mathbb {R}\)  is called the link function, \(x^* \in \mathbb {R}^d\)  is assumed to be a sparse vector and \(\langle \cdot ,\cdot \rangle \)  represents the Euclidean inner-product between two vectors. Such single-index models are widely used in statistics, machine learning and economics [1]}. A standard choices of the link function \(g\)  is the square function, in which case, the model is also called as the sparse phase retrieval model [2]}. Here, \(a\)  is the input data which is assumed to be independent of the noise \(\zeta \) . In this case, \(\xi := (a,b)\)  and the if we consider the squared-loss, then \(U(x,\xi ):= (b - (\langle a, x\rangle )^2)^2\)  and is non-convex in \(x\) . The goal is to estimate the sparse index vector \(x^*\)  in a risk-averse manner, as they are well-known to provide stable solutions [3]}. To encourage sparsity, the set \(\mathcal {X}\)  is the \(\ell _1\)  ball [4]}.
